$ E = \left[\begin{array}{rrr}5 & -2 & -1 \\ 5 & 2 & 0\end{array}\right]$ $ B = \left[\begin{array}{rr}0 & 2 \\ -1 & -2 \\ 1 & -1\end{array}\right]$ What is $ E B$ ?
Answer: Because $ E$ has dimensions $(2\times3)$ and $ B$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ E B = \left[\begin{array}{rrr}{5} & {-2} & {-1} \\ {5} & {2} & {0}\end{array}\right] \left[\begin{array}{rr}{0} & \color{#DF0030}{2} \\ {-1} & \color{#DF0030}{-2} \\ {1} & \color{#DF0030}{-1}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ E$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ E$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ E$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rr}{5}\cdot{0}+{-2}\cdot{-1}+{-1}\cdot{1} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ E$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{5}\cdot{0}+{-2}\cdot{-1}+{-1}\cdot{1} & ? \\ {5}\cdot{0}+{2}\cdot{-1}+{0}\cdot{1} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ E$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{5}\cdot{0}+{-2}\cdot{-1}+{-1}\cdot{1} & {5}\cdot\color{#DF0030}{2}+{-2}\cdot\color{#DF0030}{-2}+{-1}\cdot\color{#DF0030}{-1} \\ {5}\cdot{0}+{2}\cdot{-1}+{0}\cdot{1} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{5}\cdot{0}+{-2}\cdot{-1}+{-1}\cdot{1} & {5}\cdot\color{#DF0030}{2}+{-2}\cdot\color{#DF0030}{-2}+{-1}\cdot\color{#DF0030}{-1} \\ {5}\cdot{0}+{2}\cdot{-1}+{0}\cdot{1} & {5}\cdot\color{#DF0030}{2}+{2}\cdot\color{#DF0030}{-2}+{0}\cdot\color{#DF0030}{-1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}1 & 15 \\ -2 & 6\end{array}\right] $